##  [First-Order Theory](/first-order-theory) 

  ##  [First-Order Theory](https://puremath.quantumdictionary.io/first-order-theory-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A set of sentences expressed in first-order logic over a fixed signature, closed under logical consequence, that specifies properties intended to hold in structures of that signature.

 

 

 

 

 





 

 



 ##  [First-Order Theory](https://algebra.quantumdictionary.io/first-order-theory-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A set of first‑order sentences in a fixed signature (vocabulary) — closed or considered as a generating set for its deductive closure — whose intended models are the structures that satisfy every sentence in the set.